Optimal execution is a problem of significant importance for algorithmic traders in modern
financial markets. Here, a trader must decide an optimal trading strategy to buy (or sell) a fixed
quantity of an asset. Typically, there is a trade-off between trading quickly, which minimizes the
risk of adverse price movements, and trading slowly, which minimizes transaction costs. In their
seminal paper, [1]} framed this liquidation problem as a mean-variance optimization
problem, that optimizes a combination of the average cost (i.e., the expected implementation
shortfall) and the variability of the cost (i.e., the variance of the implementation shortfall).
In that setting, they explicitly derive the optimal liquidation schedule, which is parameterized
by the risk-aversion level of the trader. Their framework and suggested solution have become
standards in this area, serving as a starting point for more complicated models and trading
strategies, both in the academic literature and among practitioners.
